What was wrong with the old "zero" policy?
>Meanwhile, competitor countries continue to emphasize traditional, rote-like teaching and learning methodologies when it comes to math.
Rote memorization is good for a few things, like the 36 nontrivial entries on the multiplication table. But it's just not true that Singapore teaches math "by rote" in general. It is true that math teachers in East Asia are expected to know math. See this comment by jacobolus:
The relevant book:
Readers of Hacker News who have been reading the Chinese press in the original Chinese since the 1970s and who have been to the Chinese-speaking world since the 1980s and thereafter (as I have) get really sick and tired of credulous Westerners believing propaganda from China, that's all. I can remember a time from the 1980s when China (by contrast with Taiwan) was praised by a French student I knew at the time for having no air pollution. I pointed out to him then that that was because China was only just beginning to develop economically at the time. (In that year, the per-capita G.D.P. of Taiwan was ten times that of China.) China has made some interesting progress on some issues, and I often appear here on Hacker News pointing out that even the United States can learn quite a bit from how China does primary school instruction in mathematics, for example.
But China is still a dictatorship, with none of the free and fair elections or free press that Taiwan now has. (Taiwan was a dictatorship when I first lived there, but a thriving democracy the second time I lived there.) China's air pollution is now about the worst in the world--it could have put in pollution controls as it industrialized, as the technology for pollution control was well known by then, but the dictators of China chose to industrialize without minimizing pollution. In general, when I see a gee-whiz popular press story about China here on Hacker News, I simply don't believe the first submission. I look for comments in the thread by some of the Hacker News participants who currently live in China and who know the language well (the latter criterion is more stringent, but necessary) to check whether the story is plausible at all.
 In this thread, the earlier comment by a Hacker News participant who lives in China deserves an upvote.
I don't know him in person but I have emailed him. He is in touch with a lot of eminent experts (I know some of the same eminent experts and see some of the regularly in person), but he hasn't imbued their worldview about human genetics, forged after years of pursuing other worldviews that don't hold up to experimental test. But your comment is fair, so I'll recommend here for you and for onlookers some writings by people who are experts in psychometrics and human behavior genetics.
The review article
Johnson, Wendy; Turkheimer, Eric; Gottesman, Irving I.; Bouchard Jr., Thomas (2009). Beyond Heritability: Twin Studies in Behavioral Research. Current Directions in Psychological Science, 18, 4, 217-220
includes the statement "Moreover, even highly heritable traits can be strongly manipulated by the environment, so heritability has little if anything to do with controllability. For example, height is on the order of 90% heritable, yet North and South Koreans, who come from the same genetic background, presently differ in average height by a full 6 inches (Pak, 2004; Schwekendiek, 2008)."
Johnson, W. (2010). Understanding the Genetics of Intelligence: Can Height Help? Can Corn Oil?. Current Directions in Psychological Science, 19(3), 177-182
looks at some famous genetic experiments to show how little is explained by gene frequencies even in thoroughly studied populations defined by artificial selection.
"Together, however, the developmental natures of GCA [general cognitive ability] and height, the likely influences of gene-environment correlations and interactions on their developmental processes, and the potential for genetic background and environmental circumstances to release previously unexpressed genetic variation suggest that very different combinations of genes may produce identical IQs or heights or levels of any other psychological trait. And the same genes may produce very different IQs and heights against different genetic backgrounds and in different environmental circumstances. This would be especially the case if height and GCA and other psychological traits are only single facets of multifaceted traits actually under more systematic genetic regulation, such as overall body size and balance between processing capacity and stimulus reactivity. Genetic influences on individual differences in psychological characteristics are real and important but are unlikely to be straightforward and deterministic. We will understand them best through investigation of their manifestation in biological and social developmental processes."
(The review by Johnson, by the way, is rather like Tao's understanding of how mathematical talent develops in individuals, which prompted the blog post kindly submitted here.)
A comprehensive review article for social scientists on genetic research on IQ emphasizes what is still unknown.
Chabris, C. F., Hebert, B. M., Benjamin, D. J., Beauchamp, J., Cesarini, D., van der Loos, M., ... & Laibson, D. (2012). Most reported genetic associations with general intelligence are probably false positives. Psychological science, 23(11), 1314-1323. DOI: 10.1177/0956797611435528 http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3498585/
"At the time most of the results we attempted to replicate were obtained, candidate-gene studies of complex traits were commonplace in medical genetics research. Such studies are now rarely published in leading journals. Our results add IQ to the list of phenotypes that must be approached with great caution when considering published molecular genetic associations. In our view, excitement over the value of behavioral and molecular genetic studies in the social sciences should be tempered—as it has been in the medical sciences—by an appreciation that, for complex phenotypes, individual common genetic variants of the sort assayed by SNP microarrays are likely to have very small effects. Associations of candidate genes with psychological traits and other traits studied in the social sciences should be viewed as tentative until they have been replicated in multiple large samples. Doing otherwise may hamper scientific progress by proliferating potentially false results, which may then influence the research agendas of scientists who do not appreciate that the associations they take as a starting point for their efforts may not be real. And the dissemination of false results to the public risks creating an incorrect perception about the state of knowledge in the field, especially the existence of genes described as being 'for' traits on the basis of unintentionally inflated estimates of effect size and statistical significance."
The newer publications on the topic are not changing the picture significantly. If one desires to develop a child's mathematical ability, for example (a problem I have pondered four times over as a parent), then the thing to do, after gaining whatever favorable shuffle of genes one can through thoughtful choice of a marriage partner, is to ensure that the child receives a sound primary education in mathematics. That is rarely done in the United States, but it's something parents can do if they know mathematics well through some other channel, for example having lived in another country.
To answer one frequently asked question in the previous comments, no, I don't think any school anywhere includes teachers who use all these tricks in their teaching, and, yes, I think most of these classroom teacher tricks are specific to the United States. In the United States, the great majority of elementary school teachers are generalists, teaching all school subjects to their pupils, and their higher education does not prepare them well for teaching mathematics. By contrast, elementary teachers in many countries where students learn more mathematics more thoroughly are subject-matter specialists, with mathematics teachers teaching only elementary mathematics, and other teachers teaching elementary pupils other subjects.
My teaching is in two contexts: since 2007, I have taught voluntary-participation, extracurricular courses in prealgebra mathematics with additional advanced topics to middle-elementary age (mostly fourth grade) pupils in weekend supplementary classes. Many of those students are quite advanced intellectually for their age. They come to my classes (mostly through word-of-mouth recommendations from their parents' friends) from a ten-county expanse of Minnesota. Just this school year (that is, just since August) I am also on the faculty of an independent school, teaching all sixth grade pupils and "honors" seventh grade students mathematics at about the same level, although it is my intention this school year to move the seventh graders along into topics that can properly be called "beginning algebra" topics (to solving quadratic equations and graphing systems of two linear equations on the Cartesian plane).
The school where I teach is reforming its mathematics curriculum with advice from a nonprofit consulting organization. The reform program at my school is informed by international best practice in primary and secondary education and by what mathematics background is necessary to succeed in higher education in universities like MIT. (The founder of the consulting organization is an MIT alumna.) As the submitted ebook says, the hardest thing for a teacher to do is to encourage students to think rather than just rely on a mindless trick. This year I will have to set homework and tests that I write myself to probe for deep understanding of mathematical concepts and relentlessly try to find out how (and even whether) the learners in my care think about mathematics. Most of the rest of today on my weekend schedule is slated for writing a major unit test for my seventh graders, who use an excellent textbook that is part of a textbook series that doesn't provide teachers with ready-made unit tests. The textbook is based on problem-solving and explaining mathematics from first principles (Chapter 1 takes the field properties of the real numbers as an axiom system to explain many principles of arithmetic) and is the best textbook for its topic available in English.
(I liked the old website of this organization better than the new website.)
The Art of Problem Solving website is a treasure trove of interesting mathematics education resources for learners of all ages.
The point in the article that students have to be acculturated to what their school expects is very well taken. Students at a school-within-a-school for highly gifted students (a situation that exists in my local school district) have to be especially brought on board a school culture that differs from everyday school culture. And so it is for any school that introduces a change in curriculum, and for any school enrolling students who used to live elsewhere and attend other schools (a VERY common situation in the United States, and a situation I experienced while growing up).
Several comments here talk about "copying" or "regimentation," but the fine article points out that the Chinese teacher wanted to inductively lead students through a proof of the Pythagorean theorem--something I have done for much younger students here in the United States--while the British students insisted on just being told applications of the theorem without having to think about how the theorem is proved. A Minnesota Public Radio report just yesterday, based on a recent Aspen Ideas Festival discussion, "Is Math Important?" includes statements that I think are factually incorrect about comparisons between education in the United States and education in China, but I have to agree with Professor Jo Boaler's statement that a mathematics lesson in secondary school in China is anything but memorization--it is all about students discovering mathematical ideas by pursuing a few hard problems each day with group discussion. There is a whole book about what Americans don't know about how elementary mathematics is taught in China that is a good read for any participant here on Hacker News.
To answer the question posed by the article title, "Not without a lot of careful preparation, but it could possibly be an improvement for some students in some British schools."
你好. I used to live in Taiwan, and I have a big bunch of mathematics textbooks from China, as well as some recent math students in my local mathematics classes who grew up in China. Mathematics instruction in China is not at all about memorization, in the sense used in this article. A good book on the topic of mathematics instruction in the United States and China is Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma. Her field study, with sponsorship by the United States federal government, made clear that elementary mathematics instruction in China is much richer and deeper and more conceptual than in the United States.
But the idea of the article is true: When you just memorize, it makes it difficult to apply the principals that you are learning to other situations/problems.
Yes. To memorize without underlying understanding is no more useful than not memorizing at all. Developing deep understanding of a topic usually brings along with it sufficient memorization of fundamental facts that the problem-solver will be able to solve some problems on the spot, and know when to look things up if needed.
Alas, in general, American schools are underperforming in teaching the most advantaged members of American society, especially in mathematics,
while not doing well by the least advantaged young people in American society either. The current efforts in preschools reported in the interesting article submitted here should be followed up by further reform of elementary school mathematics instruction. The book Knowing and Teaching Elementary Mathematics by Liping Ma is especially informative about what to do to help all learners in the United States learn mathematics better in early schooling.
I'll go down the checklist item by item.
"Does the story compare the demographics of the student population served by charter schools to the demographics of local public schools?"
What aspect of "demographics" is most relevant here? There are already charter school programs (not many, but a few) that do better with mostly pupils who come from poor homes and parents without higher education than many other schools do with pupils who come from high-income homes. A lot of public schools in the United States serve up mediocre results while serving well off pupils.
"Does it include data on the charter school attrition rate?"
This is a fair item to put on the checklist, and should be reported more in stories about schools in general. In other words, a story about public schools should also report on their attrition rate, and on what rate at which pupils move in and out of the school district, and in general how stable or unstable enrollments are at all schools being compared. (The United States has a highly geographically mobile population, and it would be a very unusual school that has the same group of pupils enrolled by sixth grade as was enrolled in first grade five years earlier.)
"Does it include data on how the students who leave the charters compare to students who leave public schools?"
Again, this is a question well worth asking, but we shouldn't guess, until we have seen the answer to questions like this, whether charter schools or public schools would come out better in a comparison of this kind. We should be checking the facts in each individual case all around the country.
"Does it include numbers of students expelled?"
Since the early 1990s, I have been aware of homeschooling advocates who talk about "push outs" (not "drop outs") from the public school system, kids who are basically told to like it or lump it in the public school system. People leaving the public school system should always be looked at closely by researchers to figure out why they left, and whether they are later able to find a more fitting school situation in which they can learn better.
"Does it include numbers of students suspended?"
As above, this is a question to ask about any kind of school. Moreover, at the extremes, a school that never suspends any pupil is possibly a school that is not trying hard enough to maintain a learning environment for all other pupils enrolled, so a researcher would want to look at grounds for suspension, procedures surrounding suspension, what corrective steps are taken when a pupil is suspended, and so on. I've certainly seen suspension abused in public school settings.
"Does the story focus exclusively on test scores?"
This is a general defect of reporting on schools. At the broad statistical level, and especially for international comparisons of school systems, test scores are mostly what we have to look at. What I also look at, as someone who has lived in more than one country and who reads more than one language, is the actual item content of textbooks and the attitudes toward learning shown by teachers and pupils. I think some school systems in other countries compare very favorably to those in the United States (not least because pupils in those other countries begin foreign language study much earlier than pupils in the United States, a fact not usually reflected in international testing programs). Educational researchers should be looking more closely at the actual item content of school textbooks and at the specific teacher practices in each classroom.
"If so, has someone, with educational expertise, visited the school to determine if the school focuses on test prep at the expense of a rich curriculum?"
As above, we should be looking in detail at what actually goes on in the classroom. Indeed, we should be looking at whether persons employed as teachers are actually teaching anything, which we shouldn't assume sight unseen.
"Are the test scores reported outside of school assessments such as the SAT/ACT or does the story only report test scores of exams that are proctored in-house?"
This is also important, and also a question that should be applied to news stories about schools in general, not just about charter schools. So far many of the worst cheating scandals in school testing programs have originated in regular public schools.
"Does the story account for the fact that, due to the need to apply to the charter school, parents of the students at charters are, on average, likely to be more engaged in education than the parents of students at public schools?"
The policy response to this is to make school choices much more widely available to many more families. I have seen some examples of helpful reforms where I live. Minnesota, where I now live and where I grew up, has had largely equal per-capita funding for public school pupils statewide since the 1970s. The state law change that made most school funding come from general state appropriations rather than from local property taxes was called the "Minnesota miracle."
Today most funding for schools is distributed by the state government on a per-pupil enrollment basis. The funding reform in the 1970s was followed up by two further reforms in the 1980s. First, the former compulsory instruction statute in Minnesota was ruled unconstitutional in a court case involving a homeschooling family, and a new compulsory instruction statute explicitly allows more nonpublic school alternatives for families who seek those. Second, the Legislature, pushed by the then Governor, set up statewide open enrollment and the opportunity for advanced learners to attend up to two years of college while still high school students on the state's dime.
And Minnesota also has the oldest charter school statute in the United States. We haven't had big problems with charter school performance here.
Parents in Minnesota now have more power to shop than parents in most states. That gets closer to the ideal of detecting the optimum education environment for each student (by parents observing what works for each of their differing children) and giving it to them by open-enrolling in another school district (my school district has inbound open-enrollment students from forty-one other school districts of residence) or by homeschooling, or by postsecondary study at high school age, or by exercising other choices.
The educational results of Minnesota schools are well above the meager results of most United States schools, and almost competitive (but not fully competitive) with the better schools in the newly industrialized countries of east Asia and southeast Asia. It's a start. More choices would be even better.
"Does it exclusively or primarily cite reports funded by pro-charter or conservative think tanks?"
Here it's regrettable that we don't know the identity of the author of this checklist. Do we know whether or not the author is funded by pro-public-school interest groups? Disclosure of that would be helpful here.
"Does it include quotes from academic scholars or does it just cite charter school advocates?"
I can recommend a website full of the writings of actual academic scholar to Hacker News participants interested in education reform. That is the Education Next website, which is better supplied than most English-language websites about education policy with information about education policy outside the United States, which is important for a reality-check on issues discussed in this country.
"Does it identify advocates or simply call them 'experts' or 'researchers'?"
This is a funny question for a commentator who insisted on anonymity to ask.
"Does it compare the resources available to charter schools to those available to public schools?"
In Minnesota, and in all states I have checked, the resources available to charter schools (funding per student, and availability of buildings) are strictly less than the resources available to public school disticts, by law. Yes, lets's check that detail carefully.
It's commendable to receive a suggestion that we should check news stories carefully. We should always do that here on Hacker News. Let's check stories on education policy especially carefully, as public school spending is a huge part of state budgets and a lot of interest groups are lined up behind the public trough.
The author of the submitted article writes, about a seventh grade class, "The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability." As the author makes clear, the particular lesson arises from the new Common Core State Standards in mathematics, which are only recently being implemented in most (not all) states of the United States, following a period of more than a decade of "reform math" curricula that ended up not working very well. I am favorably impressed that the lesson asked students to put their numerical estimates of probability on a number line--the real number line is a fundamental model of the real number system and its ordering that historically has been much too unfamiliar for American pupils.
The author continues by elaborating on his main point: "When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one 'answer' and then call it a day." I like the author's discussion of that issue, but I think she misses one contributing causal factor--TEACHER education in the United States in elementary mathematics is so poor that most teacher editions of mathematics textbooks at all levels differ from the student editions mostly just in having the answers included and don't do anything to develop teacher readiness to respond to a different approach in a student's reasoning.
What I LOVE about the Singapore Primary Mathematics series, which I have used for homeschooling all four of my children, is that the textbooks encourage children to come up with alternative ways to solve problems and to be able to explain their reasoning to other children. The teacher support materials for those textbooks are much richer in alternative representations of problems and discussions of possible student misconceptions than typical United States mathematical instruction materials before the Common Core. Similarly, the Miquon Math materials, which I have always used to start out my children in their mathematics instruction before starting the Singapore materials, take care to encourage children to play around with different approaches to a problem and to THINK why an answer might or might not be correct. (Those materials, both of them, are very powerful for introducing the number line model of the real number system to young learners, as well as introducing rationales as well as rote procedures for common computational algorithms. I highly recommend them to all my parent friends.)
I try to counteract the "what's the correct answer" habit in my own local mathematics classes (self-selected courses in prealgebra mathematics for elementary-age learners, using the Art of Problem Solving prealgebra textbook). I happily encourage class discussion along the lines of "Here is a problem. [point to problem written on whiteboard] Does anyone have a solution? Can you show us on the whiteboard how you would solve this?" Sometimes I have two or three volunteer pupils working different solutions--which sometimes come out to different answers [smile]--at the same time. We DISCUSS what steps make mathematical sense according to the field properties of the real numbers and other rules we learn as axioms or theorems in the course, and we discuss ways to reality-check our answers for plausibility. We don't do any arithmetic with calculators in my math classes.
 When I last lived overseas, I had access to the textbook storage room of an expatriate school that used English-language textbooks from the United States, and I could borrow for long-term use surplus teacher editions of United States mathematics textbooks. They were mostly terrible, including no thoughtful discussion at all of possible student misconceptions about the lesson topics or of alternative lesson approaches--but they were all careful to show the teachers all the answers for the day's lesson in the margins next to the exercise questions.
Students don't have to do that anywhere. And they don't do that anywhere. I have lived in east Asia (and my three older children all have) and I challenge your assumption that pupils there study for fourteen hours a day (they most certainly do not) or that everything about schools there is a "pressure cooker environment." There are trade-offs involved in living anywhere, rather than living somewhere else, but you really owe it to yourself to find out more about how effective primary teaching is done before concluding that it shouldn't be tried in the United States. As they say in Chinese-speaking countries, "百聞不如一見" (hearing about something a hundred times isn't worth seeing it once). I offer for your reading pleasure a reading suggestion, full of food for thought, namely Liping Ma's book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. Reviews of this book appear below the book link.
Anyway, we can get other reality checks on this issue by seeing what people who have studied the United States school curriculum say about secondary schools here, for example "What’s Holding Back American Teenagers? Our high schools are a disaster." The last federal sample survey on the issue resulted in news reports such as, "School is too easy, students report," suggesting that a lot of learners here don't appreciate dumbed-down school lessons that waste their time without providing a return in new knowledge and skills.
a) OBSERVE what the school day looks like in other countries,
b) analyze textbooks and teaching practices that produce superior results
c) take careful note of classroom practices that respect learners and help learners learn deeply and not just superficially,
d) systematically train teachers in effective teaching techniques.
Boys and girls, all over the world, are exposed to many different kinds of school schedules and school practices, and it is the basic procedure of educational science to carefully observe the different combinations of home, societal, and school influences on children and try to figure out what helps learners have the best results.
Better teaching makes for better results in learning.
I have read some of the curriculum standards adopted in various states over the last decade and have examined the item content of some of the No Child Left Behind Act state tests implemented during the same period. The curricula were often quite lousy, and the tests rather poorly constructed. But neither so constrained teachers that we can conclude that they made things WORSE for teachers than before the Act and the associated tests. Teachers are in the classroom to help pupils and students learn something. Defining part of what that something is by no means prevents teachers from teaching more. A teacher who self-educates about good quality research on human learning
and about effective teaching
can help learners learn better even if the surrounding pattern of school regulation is less than ideal.
I am a teacher of prealgebra-level mathematics in private practice. (In earlier years I was a classroom teacher of English as a second language or of Chinese as a second language.) My elementary-age pupils come to me for lessons after attending their regular school lessons each week. All my clients have to pay me (my nonprofit program also offers financial aid, up to a full fee waiver, for families with financial need) after already paying their taxes for my state's friendly public schools, and some of my clients come to my program after paying out of pocket for a privately operated classroom school or as a supplement to family homeschooling. I don't give my pupils letter grades, and tests I offer to the pupils are from national voluntary participation mathematics contests, which they take (or not) as one of several reality checks on how they are learning the course material. Parents from a wide variety of school districts have told me that their children do much better on various kinds of school tests after taking my course, even though my course is explicitly NOT test-prep, and even though I don't align my curriculum to the curriculum presupposed by any testing program.
Children who learn how to use their brains to think
can handle novel problems and are not afraid of tests. Children who are overprotected in school from learning challenges outside the standard curriculum often get scared and shut down when tested, even when tested on the curriculum content they have studied over and over. I'm all about helping young learners be unafraid to take on challenges. If a teacher is not doing that, what is the teacher doing?
It's probably worth noting for other HN participants that the blog from which this guest post was submitted has had guest posts before that many Hacker News readers caught omitting many of the key facts of the described situation,
until that hiding the ball was outed by more thorough bloggers who checked the facts.
AFTER EDIT: btilly kindly asks, in the first reply to this comment, what class size I teach. The class size I teach is lower than the typical class size at the schools of regular enrollment of the pupils I teach, and more to btilly's point, my total enrollment of students at a given time is less than the typical student load of a full-time teacher in the local public schools. That's a fair contrast between my situation and theirs. On the other hand, for the first several years of my program I was writing the whole curriculum from the ground up (as no suitable textbooks were avaiable from United States publishers) and sometimes gathering materials from three different countries just to put a lesson plan together.
More to the point of teaching large classes, it has been done and done well in many parts of the world. When my wife was growing up in Taiwan, the typical elementary school class size was sixty (60) pupils. An unusually small class would have only fifty (50) pupils enrolled. The differences in school staffing practices and teacher training to make that possible are described in book-length works
but boil down to letting classes be extra large, so that teachers can be scheduled to have joint prep time together each day in which new teachers learn from master teachers and plan lessons together. My teaching would be better if my program were big enough that I had a colleague to confer with each week, or especially each day.
and from time to time regret the gaps in their own mathematical education.
"The teachers are eager and able to learn. I vividly remember one summer class when I taught why the multiplication algorithm works for two-digit numbers using base ten blocks. I have no difficulty doing this with third graders, but this particular class was all elementary school teachers. At the end of the half hour, one third-grade teacher raised her hand. 'Why wasn’t I told this secret before?' she demanded. It was one of those rare speechless moments for Pat Kenschaft. In the quiet that ensued, the teacher stood up.
"'Did you know this secret before?' she asked the person nearest her. She shook her head. 'Did you know this secret before?' the inquirer persisted, walking around the class. 'Did you know this secret before?' she kept asking. Everyone shook her or his head. She whirled around and looked at me with fury in her eyes. 'Why wasn’t I taught this before? I’ve been teaching third grade for thirty years. If I had been taught this thirty years ago, I could have been such a better teacher!!!'"
The last time I posted a link to this article on HN, another HN participant kindly posted a link to what is surely the "secret" referred to by the elementary school teacher,
pedagogical content knowledge that would be very routine for any elementary mathematics teacher in east Asia.
(book link above, review links below)
So this advice for parents is good in helping parents provide a supportive environment for their children's mathematics learning.
I have frequent occasion to write about mathematics education here on Hacker News. My occupation is 1) providing supplemental lessons in advanced mathematics to pupils from ten counties in Minnesota through a nonprofit corporation I cofounded and 2) coordinating parent workshops and other aspects of the summer program Epsilon Camp,
perhaps the most advanced mathematics program of its kind for YOUNG learners in North America.
To date, I recommend to my own children and to my clients in my own supplemental mathematics education program that they also turn to ALEKS,
which is a commerical online site (in which I have no economic interest) delivering personalized instruction in mathematics through precalculus mathematics. The ALEKS website includes links to research publicatoins on which ALEKS is based.
I also recommend the Art of Problem Solving (AoPS)
(where I first took on the screenname that I also use here on HN) for more online mathematics instruction resources, and I also share specific links to specialized sites on particular topics with clients and with my children. I should note for onlookers that the articles on mathematics learning on the AoPS website
are very good indeed, especially "The Calculus Trap."
My children make quite a bit of voluntary use of Khan Academy (both watching videos and working online exercises) and I am gratified that my previous suggestions to the Khan Academy developers here on HN
have been followed up as Khan Academy developers have communicated with me by email about new problem formats available in their online exercises, which are becoming increasingly challenging.
Besides that, I fill my house with books about mathematics, and circulate other books about mathematics frequently from various local libraries.
I also recommend that all my students use the American Mathematics Competition
materials and other mathematical contest materials as a reality check on how well they are learning mathematics.
In general, I think mathematics is much too important a subject to be single-sourced from any source. Especially, mathematics is much too important to be left to the United States public school system in its current condition. I was rereading The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999) last month. It reminded me of facts I had already learned from other sources, including living overseas for two three-year stays in east Asia.
"Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x
"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77
Plenty of authors, including some who should be better known and mentioned more often by HN participants, have had plenty of thoughtful things to say about ways in which United States mathematical education could improve.
A discussion of the Common Core Standards in Mathematics, "The Common Core Math StandardsAre they a step forward or backward?"
gets into further details of how mathematicians look at the general school curriculum in the United States. It is not the worst curriculum possible, and survivors of the system often have access to outside resources to supplement school lessons, but the public school instruction in mathematics in the United States still shows plenty of room for improvement.
The last time I posted about these issues, a reply asked what I think about essay "Lockhart's Lament." I think it is an interesting read, but less practical for reforming mathematics education than I had hoped. I wonder if Lockhart's forthcoming book Measurement
will be a successful attempt to teach mathematical reasoning to students who have already lost confidence in learning mathematics, which would be a great contribution to society.
The last time I saw a comment of this nature in a thread on the subject of mathematics education in the United States and east Asia (both places I have lived), someone advised me not to feed the troll. But that was a different troll, and taking your comment, even where you incorrectly say "highly misleading" about my comment, as an attempt to advance the discussion, I'll invite onlookers to look at the evidence.
I backed up my statement with a link
and anyone who takes a look at Exhibit 1.1 of that link (on pages 34 and 35 of the .PDF document), which is a good example of a comparative data distribution display, can see how the national median level of performance in the United States compares to the bottom quartile level for Singapore, and on the other hand where the top quartile line for the United States appears compared to the median line for Singapore. Q.E.D.
Unfortunately, once upon a time a blogger ignorant of the large body of research on textbook content and classroom practice in different countries for elementary mathematics in different countries of the world
took the lazy way out and said that if "race" is taken into account, then the United States is second to none in provision of public education, which is simply a lie. That meme has spread through some politically tendentious blog networks, but every serious professional researcher on comparative education policy can, and does, point to more meaningful differences between the United States and other countries. It would have helped that blogger also to be more familiar with the huge literature on "race" issues in countries all over the world,
but let me just disagree with the suggestion in your comment by pointing that nobody who makes the suggestion made by the blogger has actually gathered the data to show all the steps to prove that "race" as such makes any difference at all in educational attainment. Meanwhile I have taken care, in links already shown in my first comment above to document both the known inferiority of provision of primary education to some "race"-defined groups in the United States
and the degree to which other countries outperform the United States in providing primary education to the most disadvantaged groups in each of those countries.
Moreover, and this link is new to this thread, but not newly posted to Hacker News,
the United States is conspicuous in how little it meets the educational needs of its strongest students in mathematics.
in the big picture it's hard to conclude that our education is failing
There is certainly room for semantic disagreement about how bad performance has to be before it is regarded as "failing" performance, but I note for the record that the United States has abundant resources devoted to K-12 schooling
but underperforms compared to what other countries do with less abundant resources. I didn't use the word "fail" or "failing" or "failure" in my comment, but I did suggest, and I think I suggested this with warrant, that United States schools could do a better job of teaching fraction arithmetic to the young people in their care.
Any competent music teacher (I write as the husband of a piano teacher) can perform music with musical expression, and thus show students examples of the beauty of music. But the very best music teachers are also intimately familiar with all the isolated subskills that build into understanding a piece of music, and controlling the performer's muscle movements, and responding to the audience in a live setting to build a coherent, musically expressive performance. My wife teaches skills such as "music mapping,"
proper hand position,
and how to tie those and many other skills together
as part of a comprehensive process of teaching making music.
The basic problem with mathematics education at the elementary school level in the United States (see my previous reply to this thread
generally commenting on the submitted article) is that elementary school mathematics teachers can do NONE of the comparable things with mathematics that a good music teacher can do with music. They cannot isolate and focus on useful techniques, they cannot put on an example performance of solving an interesting, challenging problem, and they cannot make connections between their (poor) understanding of the problems found in elementary mathematics and their students' (often better, but different) understanding of the same problems. In the United States, people mostly seek music instruction for young learners from the private enterprise system. My wife gains most of her new clients, who are crammed into her busy teaching schedule as previous clients graduate from secondary school and go off to university, from friends of current clients who are happy with her work. By contrast, an elementary school teacher in a typical government-run school in the United States teaches on a take-it-or-leave-it basis, with attendance being compulsory in default of a government-approved alternative, and funding guaranteed to the school, and thus employment guaranteed to the teacher, whether the learners learn mathematics or not. Systemic change is necessary to get mathematics taught as music is taught to elementary-age pupils in the United States.
For an eye-opening look at how elementary mathematics teachers could be prepared, and how that would help learners, see Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.
That book is a very enjoyable--but rather shocking--read, full of information about how to teach mathematics for "Profound Understanding of Fundamental Mathematics."
In the West and in the East, there has always been wisdom contrary to the conventional wisdom that small class sizes are always better. Indeed, the Roman author Quintilian, writing about rhetoric, derided teachers who insisted on small class sizes. In Quintilian's view, the true test of a teacher was being able to engage and enlighten a large class. Quintilian described teachers who could only handle small class sizes as no better than baby-sitting slaves. He wrote, "all good teachers like a large class and think they deserve a bigger stage" while it is the "weaker teachers, conscious of their own defects, who cling to individual pupils and seem content" (Book I of his Institutio Oratoria).
East Asian schools, which are still plainly superior to those of the United States in the view of informed observers,
have characteristically large class sizes, the better to ensure that teachers are more stringently selected and that they have work hours during the school day to confer with master teachers of their subject. More details of how schools are organized in some of the conspicuously successful countries can be found in
That's how I read it too. And that is a finding of international studies of primary mathematics education. The newly industrialized countries of east Asia do much better in primary mathematics education, using textbooks (and teacher professional development to ensure that the textbooks are well applied in the classroom) such that key concepts are presented from multiple points of view, to reach as many learners as possible. The TIMSS test results
suggest this is working out well for those countries. A book-length description of how mathematics primary education differs between much of the United States and much of China
is an eye-opening read.
(By the way, the scatter of data points around the regression line in their plot suggests that the model is subject to large degrees of error in prediction.) It would take an experimental design (randomly assigning one group of teachers in the same country to receive pay raises while another group does not, with before-and-after comparisons of pupil performance) to show that paying teachers more results in higher pupil performance.
There have been hundreds of studies of educational interventions over the years,
and many thoughtful international comparisons of teaching practice,
but none of those conclude that simply raising teacher pay, without changing teaching practices and perhaps also the composition of the teaching workforce, will have much to do with raising pupil performance in any place. Raising teacher pay systematically has been tried in the United States (notably in the state of Connecticut) and has not been shown to markedly raise pupil performance.
An economist who closely studies education policy has suggested that pay and other incentives be used to encourage the least effective teachers to seek other occupations while rewarding the most effective teachers with increased compensation and more professional support.
Such a policy, he estimates (showing his work in his article) would raise United States educational achievement to the level of the highest-performing countries. This is something worth verifying by experiment, although that will be politically difficult in any state of the United States
and perhaps in Britain as well.
P.S. I'm curious about why the United States underperforms so much compared to salaries paid to teachers in the chart shown in the submitted blog post.
I agree with some points in your reply. I don't think China as a whole is well represented by the schools in its most developed urban areas. The results from Shanghai in the most recently announced test to include Shanghai surely don't reflect what students from rural areas in China would do on the same test. But even agreeing with that point, I wonder if you've had a chance to take a look at what Ma's book
says about differing classroom practices and differing lesson content between the United States and China. China is very, very, very much poorer than the United States because of the lousy policies it had in the 1950s and 1960s. But its educational policies since the 1970s have been on an increasingly sound basis, and seem to be producing admirable results in economic growth with remarkably low school budgets. But please note that I never appeal to China as a country with country-wide results that are uniformly better than those of the United States. China is especially doing well on a resources-adjusted basis, while Singapore, Taiwan, and some other countries are just plain doing well nationwide, period. (I am most familiar with Taiwan, from much time living there.)
I also agree with the idea that it's important to look at education studies "with a critical eye" and it was with that in mind that I referred fellow participants on HN on several earlier occasions to the studies showing that United States schools are underserving the most able learners,
missing opportunities to reach the top end of mathematics achievement reached by other countries. "Data doesn't lie, but analysis is often wrong and/or exaggerated," I agree, and what I find is that some forms of analysis are not even attempted by many commentators on education policy. I think writings that are good examples of good analysis
are food for thought for those of us participating on Hacker News who seek ways to improve education wherever we live.
That's a very good question. This question has not been studied as rigorously as it should have been. Here are some suggestive observations. Professor W. Stephen Wilson surveyed many colleagues (other research mathematicians) once to ask if they thought advanced mathematics could be learned without a basic understanding of arithmetic. The responses he received
included comments such as "I am shocked that there is any issue here" and "That it is even slightly in doubt is strong evidence of very distorted curriculum decisions" and "One of my favorite attacks is that we are _helping_ the students by insisting that they do things by hand because otherwise they can waste a lot of time when the calculator would fail them." One especially thoughtful comment, by a mathematician who has long thought deeply about teaching mathematics, was "It might be argued that we do not really require students to fiercely add, subtract, multiply and divide in our university courses - which is true. But we do require an automatic understanding of these operations and why they work because WE BUILD FROM THERE."
The longer story about understanding arithmetic--REALLY understanding it--and how that relates to learning beyond arithmetic can be found in the book Knowing and Teaching Elementary Mathematics by Liping Ma
(well reviewed by two mathematicians who study mathematics education)
A classic article on the subject is "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education"
A recent effort to embody strong conceptual understanding of basic skills into a mathematics textbook is Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana,
which points to what young students should be able to do WITHOUT a calculator if they really understand mathematics well.
One example I know, related to me by an economics professor, is teaching a lecture on economics in which the professor (the one who told me this story) said, as part of a calculation, "20 percent," and then was interrupted by a student who asked, "You just said '20 percent,' but you wrote '.2' on the blackboard. Why did you do that?"
Where my wife grew up, it took one teacher to teach 60 pupils. A class size of 50 was an exceptionally small class size. Several of the countries that best the United States in academic achievement
have much higher class sizes per teacher than the United States has. It is definitely possible to improve teacher productivity over the low level maintained in the United States. There are whole books on the subject.
What are the most crucial differences that United States policy-makers should be aware of as they try to improve education in the United States? (For that matter, what should east Asian policy-makers learn from the United States?) I like the books by Liping Ma
James Stigler and James Hiebert
as examples of what the United States could learn from the practices of other countries, but perhaps you have other suggestions for readers here.
which she has found very helpful.
As for mathematics, the subject I teach now, I have always cherished visual representations of mathematical concepts, for example those found in W. W. Sawyer's book Vision in Elementary Mathematics
But other mathematicians who taught higher mathematics, for example Serge Lang, recommended memorizing some patterns of multiplying polynomials by oral recitation, just like reciting a poem.
The acclaimed books on Calculus by Michael Spivak
and Tom Apostol
are acclaimed in large part because they use both well-chosen diagrams and meticulously rewritten words to deepen a student's acquaintance with calculus, related elementary calculus concepts to the more advanced concepts of real analysis.
Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma
demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.
Elementary school teachers having a poor grasp of mathematics and thus not helping their pupils prepare for more advanced study of mathematics continues to be an ongoing problem in the United States.
In light of recent HN threads about Khan Academy,
I wonder what Khan Academy users who also have read the submitted blog post by Cal Newport think about how well students using Khan Academy as a learning tool can follow Newport's advice to gain insight into a subject. Is Khan Academy enough, or does it need to be supplemented with something else?
That's a mistake of the system in the United States. In many other countries, teachers specialize by subject in the elementary grades, the better to teach their subject effectively. See Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States
for a detailed discussion of elementary math teaching, or The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom
for a broader perspective on other ways to organize schools.
Definitely baseball statistics for that audience. Also finishing times for sprints to thousandths of a second.
Not to answer your question, but to suggest useful resources for elementary math teaching, I'll suggest
an essential book for someone in your profession, and
a book with wonderful teaching tips, and
a very excellent set of exercises for any elementary mathematics teacher preparing lessons, and a great guide to the best available series of mathematics textbooks in English.
You might also try asking your question on the appropriate forum
(the forum is really for middle school math, but you could give it a try) on the Art of Problem Solving site, a generally good site for discussion of math.
(or its review by mathematician Richard Askey)
for example of ways American primary education could do better.
to a very informative book about math education and how it can be better. I strongly suspect the brain reorganization happens better and faster for learners with good math teachers rather than lousy math teachers.
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